capacitor - web.northeastern.edudielectric constants for some materials induced charge and...
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Capacitor
Any two pieces of metal (conductor)
brought near each other.
Parallel plate capacitor
L
W
Area A = L x W
Charged capacitor
“Charged Capacitor” has no net charge.
+++++++++++++++++++++++
-----------------------------------------
+Q
-Q
E=0
E=0
E
Electric Field (treating plates as
infinite) and having area A.
⎟⎠
⎞⎜⎝
⎛===A
QEE
00
1||
εεσv
d
Capacitance
+++++++++++++++++++++++++++
----------------------------------------------
+Q
-Q
⎟⎠
⎞⎜⎝
⎛==A
QE
00
1
εεσ
EdrdEVV =⋅=∆= ∫v
v
)( 0 dA
QV
ε=→
We define “capacitance”
V
QC ≡
For parallel plate capacitor
d
AC 0ε≡
d
Capacitance
V
QC ≡
Can think of capacitor as a device for storing charge
(and also energy).
Units [Capacitance] = [Coulomb/Volt] = [Farad]
L
L
d
A parallel-plate capacitor has square plates of edge
length L, separated by a distance d.
If we doubled the dimension L and halve the dimension
d, by what factor have we changed the capacitance?
A) 1.0
B) 2.0
C) 4.0
D) 8.0
E) 16.0( )22 2oo o o
2LA L LC 8
d d d / 2 d
εε ε ε= = → =
CPS Question
How big is a one Farad Capacitor?
CPS question
Assume you have two parallel plates that are separated by 1 mm.
If you estimate ε0
~ 10-11, what is the area of the plates to have a
1.0 Farad capacitance?
A) 100,000,000 square meters
B) 1,000 square meters
C) 1 square meters
D) 0.1 square meters
E) 0.001 square meters
d
AC 0ε≡
Commercial capacitors
Micro-farad capacitors in small packages are
made by making d very small ~ atomic
dimensions.
Energy Storage
Capacitor stores both charge and energy.
Suppose the capacitor is not fully charged. How much work is
required to move a little bit +dq across?
+++++++++++++++++++++++++++
----------------------------------------------
+Q
-Q
dqVdqdUdWWork ===
dqC
qdU ⎟
⎠
⎞⎜⎝
⎛=
C
Qqd
C
qdUU
Q
2
2
0
=⎟⎠
⎞⎜⎝
⎛== ∫∫
Each next piece +dq gets harder
and harder.
QVCVC
QU
2
1
2
1
2
1 22
===Potential Energy stored in the
capacitor.
CPS Question
A parallel plate capacitor is charge up (+Q on one plate and
–Q on the other). The plates are isolated so the charge Q
cannot change. The plates are then pulled apart so that the
plate separation d increases. The total electrostatic energy
stored in the capacitor?
A) Increases
B) Decreases
C) Remains constant
+++++++++++++++++++++++
-----------------------------------------
+Q
-Q
Ed QVCVC
QU
2
1
2
1
2
1 22
===
Where is this energy U?
Answer = the electric field has energy.
Energy density of E-field = uVol
U
volume
energy ==.
( )dA
Edd
A
Vol
CV
⋅
⎟⎠
⎞⎜⎝
⎛
==
202
21
.21 ε
202
1Eε=
Turns out work to charge capacitor is equal to energy stored.
Electric Field energy
This is valid for any electric field configuration in vacuum:
“empty” space contains energy!!!
Spring analogy
+++++++++++++++++++++++++++
----------------------------------------------
+Q
-Q
21
2
1Q
CU ⎟
⎠
⎞⎜⎝
⎛=2
2
1kxU =
Electric field energy Elastic energy
What is this device?
Applications
Applications (cont)
Capacitors are good for storing large amounts of energy, which
can then be accessed quickly (e.g. camera flash).
Capacitors are also ideal transducers.
Devices that covert physical quantities into electrical
signals (e.g. computer keyboard, elevator buttons).
A
d d
AC 0ε≡
Circuits with capacitors
Circuit symbol for a capacitor
More than one capacitor?...
Capacitors in parallel
( )
21
21
2121
2211
21
;
CCC
CCV
Q
VCCQQQ
VCQVCQ
VVVV
eq
total
ab
+=→
+=→
+=+===→
===
Capacitors in parallel (cont)
Think of two capacitors in parallel as adding the
area of the two capacitors
...321 +++= CCCCeq
Many capacitors in parallel:
Capacitors in series
21
2121
22
11
21
111
11
;
CCCQ
V
CCQVVVV
C
QVV
C
QVV
eq
ab
cbac
+==→
⎟⎟⎠
⎞⎜⎜⎝
⎛+=+==
====→
=
Capacitors in series (cont)
21
21
111
or
111
CC
C
CCC
eq
eq
+=
+=
...1111
321
+++=CCCCeq
Many capacitors in series:
19
CPS question
A bank of three capacitors (C1, C
2, C
3) is connected as follows.
C1=2 F
C3=1 F
C2=1 F
What is the effective total
capacitance of the bank?
A) C = 0 Farad
B) C = 1 Farad
C) C = 1.5 Farad
D) C = 2.0 Farad
E) C = 0.5 Farad
DielectricsWhat if the gap between the capacitor plates is not empty?
Dielectrics (cont)
0/VQCwithout =
withoutCC
VV
VQC
>→<
=
0but
/
Without dielectric:
With dielectric:
Increases the capacitance!!!
Dielectric constant
Dielectric constant:
V
V
C
CK 0
0
==
K
VV 0=
With the dielectric present, the
potential difference for a given charge Qis reduced by a factor K
Dielectric constants for some materials
Induced charge and polarization
PolarizationWe assume that the induced surface charge
density is directly proportional to the electric
field magnitude in the material
Without dielectric:
00 ε
σ=E
0
0
εσσ induced
K
EE
−==With dielectric:
⎟⎠
⎞⎜⎝
⎛ −=Kinduced
11σσFrom these we
obtain:
Permittivity
0εε K=
We have found:
εσ=E
0
0
εσ
KK
EE ==
Defining:
We obtain:
permittivity
and
220
00
2
1
2
1EEKu
d
A
d
AKKCC
εε
εε
==
===Capacitance
Energy density
Easy rule of thumb: replace ε by ε0
in all the equations